Extracting an $\mathbb{N}$-filtered differential modality from a differential modality
For researchers in categorical logic and differential linear logic, this result offers a systematic way to extract filtered structures from existing differential modalities, though it is an incremental theoretical extension.
The authors show that every differential modality on an additive symmetric monoidal category can be refined into an N-filtered differential modality, where morphisms correspond to polynomial maps of bounded degree. This provides a categorical framework for polynomial functions.
A differential modality is a comonad on an additive symmetric monoidal category $(\mathsf{C},\otimes,I)$, whose underlying functor we denote $!\colon\mathsf{C} \rightarrow \mathsf{C}$, together with some additional structure including a differential operator $\partial\colon!A \otimes A \rightarrow !A$. A morphism $f\colon !A \rightarrow B$ is interpreted as a smooth function from $A$ to $B$. The notion of an $\mathbb{N}$-filtered differential modality is a variant in which a notion of degree is present. Instead of a single functor $!\colon \mathsf{C} \rightarrow \mathsf{C}$, we ask for a family of functors $!_{\le n}\colon\mathsf{C} \rightarrow \mathsf{C}$ where $n \in \mathbb{N}$. Now, a morphism $f\colon !_{\le n} A \rightarrow B$ is interpreted as a smooth function from $A$ to $B$, with degree less than $n$ for some notion of degree. We prove that under mild conditions, every differential modality on an additive symmetric monoidal category with underlying functor $!\colon \mathsf{C} \rightarrow \mathsf{C}$ yields an $\mathbb{N}$-filtered differential modality with underlying functors $!_{\le n}\colon\mathsf{C} \rightarrow \mathsf{C}$. A morphism $f\colon !_{\le n}A \rightarrow B$ corresponds to a polynomial map of degree less than $n$ from $A$ to $B$, in the sense that the $(n+1)$-th derivative of $f$ is $0$.